We knew Tina Fey could do no wrong, but for a while her new show looked like it might test our faith. Awkwardly named, ditched by its original network and trailed with a promo that made it look sickly and annoying, Unbreakable Kimmy Schmidt arrived online last Friday without, in the UK at least, too much hoo-hah.

What a relief, then, to find that it’s just as smart as Fey’s other work, but with new warmth and optimism. It’s a brighter, softer, gayer 30 Rock. Created by Fey with former 30 Rock showrunner Robert Carlock and starring the wondrous Ellie Kemper (The Office), Unbreakable Kimmy Schmidt already looks like it can join Orange Is the New Black, House of Cards and Better Call Saul in the handful of shows that make Netflix essential.

Unbreakable Kimmy Schmidt: Tina Fey’s joyous new creation

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Kimmy emerges after being imprisoned for 15 years by a man who told her and three other “wives” that there had been a nuclear apocalypse outside their underground bunker. She decides to start afresh in New York, keeping her ordeal a secret to avoid becoming a gawped-at minor celeb in her Indiana home town. Happenstance gifts her a job nannying for bonkers Upper West Side mother Jacqueline (Jane Krakowski), and a flatshare with failed but fabulous actor Titus (Tituss Burgess).

Too contrived? Actually this setup frees Unbreakable Kimmy Schmidt from some of the awkwardness of a naive, sunny Midwesterner butting up against the cynicism of the city. There’s no work to be done to convince us that Kimmy doesn’t know how the big bad world operates. Plus, bonus gags about someone who has woken up in the future! Always funny. (Kimmy to Titus, telling him his dreams of stardom will come true: “You’re going to sing at the Grammys with Whitney Houston and Michael Jackson!”)

Very quickly, Kimmy – a close cousin of Leslie Knope from Parks & Rec – becomes a magic friend we should all strive to be a bit more like, starting with her perky spring wardrobe full of happy lemon cardigans, hilarious fuchsia trousers and cute floral minis. More profoundly, her efforts not to be recognised as “one of the molewomen” caricature the fear everyone has of Being Found Out, and the struggle not to be weighed down by bad stuff in your past.

In an over-connected world where everyone has heard of everything, and if they haven’t they pretend that’s not the case, and in any case, it’s OK they’ve looked it up on their phone now, Kimmy strides unembarrassed into each new day and says: I really have no idea what’s going on here – but how bad can it be? Every episode brings a new opponent telling her she doesn’t fit in or can’t do what she plans. They are always wrong. That word “unbreakable” is not in the title by accident.

But this is a Tina Fey production, so undercutting the primary-coloured, heel-clicking New Girl quirk is bitter sharpness; an acute awareness that people out there tend to be mean, aggressive and corrupt, even if we’re not going to let them come in here.

Most of the show’s many, many fine gags come out of that 30 Rock trick of people constantly revealing the mad or bad things they’ve seen. Kimmy remembers life in the bunker (“I’ve gotten botulism a bunch of times from spoiled canned goods, so I know lots of ways to make people throw up”), Titus recalls thespian fiascos (“When I first moved here, they were doing an all-black production of Oklahoma! called Alabama!”), and Jacqueline offers glimpses into her inhumanly pampered lifestyle (“In preparation for Julian’s return, I went to the gyno-dermatologist. I need to lie down with my feet and heart above my vagina”).

Season two is on order – and you can envisage the show running and running. Inexplicably dropped by NBC, the eminently bingeworthy Unbreakable Kimmy Schmidt feels more at home on Netflix, where the whole-series dump lets you step into a show’s world when you like, and live there for a while. Kimmy’s world is a place of joy.